Superconductivity in Three-Dimensional Spin-Orbit Coupled Semimetals

Motivated by the experimental detection of superconductivity in the low-carrier density half-Heusler compound YPtBi, we study the pairing instabilities of three-dimensional strongly spin-orbit coupled semimetals with a quadratic band touching point. In these semimetals the electronic
structure at the Fermi energy is described by spin j = 3/2 quasiparticles, which are fundamentally different from those in ordinary metals with spin j = 1/2 . We classify both local and nonlocal pairing channels in j = 3/2 materials and develop a general approach to analyzing pairing instabilities therein, thereby providing the computational tools needed to investigate the physics of these systems beyond phenomenological considerations. Furthermore, applying our method to a generic density-density interaction, we establish that: (i) The pairing strengths in the different symmetry channels uniquely encode the j = 3/2 nature of the Fermi surface band structure---a manifestation of the fundamental difference with ordinary metals. (ii) The leading pairing instabilities are different for electron doping and hole doping. Finally, we argue that "polar phonons," i.e. Coulomb interactions mediated by the long-ranged electric polarization of the optical phonon modes, provide a coupling strength large enough to account for a Kelvin-range transition temperature in the s-wave channel, and are likely to play an important role in the overall attraction in non-s-wave channels. Moreover, the explicit calculation of the coupling strengths allows us to conclude that the largest two non-s-wave contributions in YPtBi occur in non-local channels, in contrast with what has been commonly assumed.